#8393323

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,5)(3,7,8,6)(9,10,15,17)(11,18,12,16)(13,21,23,19)(14,20,24,22)(25,26)(27,28)(29,30)(31,33)(32,34)(35,36)(37,38)(39,40);(2,5)(6,7)(9,11,15,12)(10,16,17,18)(13,22,23,20)(14,21,24,19)(25,27)(26,29)(28,34)(30,32)(31,35)(33,37)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,17)(14,24)(16,18)(19,21)(25,28,32,29)(26,27,34,30)(31,37,39,36)(33,38,40,35);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,18,17,16)(13,20,23,22)(14,19,24,21)(25,29,32,28)(26,27,34,30)(31,37,39,36)(33,35,40,38);(1,3,4,8)(2,6,5,7)(9,13,15,23)(10,19,17,21)(11,20,12,22)(14,16,24,18)(25,30,32,27)(26,29,34,28)(31,38,39,35)(33,37,40,36);(9,14)(10,20)(11,21)(12,19)(13,16)(15,24)(17,22)(18,23)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300519);